This type of device is known in more than one publication of prior art (see, e.g., West German Laid-Open Application No. 1,959,679, West German Patent Specification No. 2,317,750, West German Laid-Open Patent Application No. 3,304,951). In the devices of known construction, the degassing element is a flat membrane (West German Patent Specification No. 2,317,750) or at least part of the casing (West German Laid-open Patent Application No. 3,304,951). As a rule, these devices are attached to the arm of the patient, as a result of which the casing, though it can no longer be rotated about the longitudinal axis, can be brought into an inclined position by the arm movement of the patient. In this respect, viewed in the longitudinal direction of the casing, the degassing element must be disposed at both ends of, and in the area of or above, the longitudinal axis of the casing so as to produce a reliable separation of the gas bubbles from the fluid in any inclined position of the casing. All the devices of known construction have the disadvantage that, owing to the required porosity of the degassing elements, the latter are not transparent, and it is therefore impossible to check whether the gas bubbles are being removed from the fluid. Moreover, when using flat membranes as the degassing element, extra work is necessary to tightly embed the degassing elements during manufacture, which results in greater expense for the known devices. This has particularly unfavorable consequences, since these devices are disposable articles and are therefore mass-produced. Since these devices are usually attached with adhesive plaster to the patient's arm, the danger exists that the outlet opening for the gas, frequently formed as an opening in the casing wall facing away from the patient's arm, will be covered by the adhesive plaster, thereby preventing effective degassing of the fluid.